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Chapter 1: Problem 23

Find an equation of the given line. Parallel to \(y=3 x+7 ; x\)-intercept is 2

Short Answer

Expert verified
y = 3x - 6

Step by step solution

01

Identify the slope

The given line is parallel to the line with equation y = 3x + 7. Since the lines are parallel, they will have the same slope. Thus, the slope (m) of the new line is 3.
02

Use the x-intercept to find the y-intercept

The x-intercept of the new line is given as 2. This means when x = 2, y = 0. Using the slope-intercept form of the line equation, y = mx + b, substitute x = 2, y = 0, and m = 3 to find the y-intercept (b). 0 = 3(2) + b 0 = 6 + b b = -6
03

Write the equation of the line

Now that the slope (m) is 3 and the y-intercept (b) is -6, substitute these values into the slope-intercept form equation y = mx + b. This gives us the equation of the line: y = 3x - 6

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallel Lines
Parallel lines are lines in a plane that never intersect. They always have the same slope. In our example, we are given a line with the equation \(y = 3x + 7\). The slope of this line is 3. Since the lines are parallel, the slope of our new line will also be 3. If you remember one thing about parallel lines, it's that their slopes are equal. This is a key property that helps in finding the equation of a line parallel to another.
Slope-Intercept Form
The slope-intercept form of a line is an easy way to write the equation of any straight line. It is written as \( y = mx + b \). Here, \(m\) is the slope of the line, and \(b\) is the y-intercept. In our problem, we know the slope \(m\) is 3 because the line is parallel to \(y=3x+7\). To find \(b\), the y-intercept, we use the given x-intercept of 2. Using the formula, we substitute \(x = 2\) and \(y = 0\) to solve for \(b\). This gives us the equation: 0 = 3(2) + b b = -6Therefore, the equation of our line becomes \(y = 3x - 6\).
X-Intercept
The x-intercept is the point where the line crosses the x-axis. At this point, the value of y is always zero. In our example, the x-intercept is given as 2, meaning the line crosses the x-axis at (2,0). This is a crucial piece of information that helps us find the y-intercept (b) in the slope-intercept form of the line. By plugging \(x=2\) and \(y=0\) into the equation \(y = 3x + b\), we can solve for \(b\), helping us fully define the equation of the line.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. Here, the value of x is always zero. For the equation of our line, we have already found that \(b=-6\), meaning the line crosses the y-axis at (0, -6). This is an important characteristic that allows us to understand the position and behavior of the line on the graph. Combining the slope (3) and the y-intercept (-6), we derive the final line equation \(y = 3x - 6\).

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Most popular questions from this chapter

Let \(f(x)\) be the value in dollars of one share of a company \(x\) days since the company went public. (a) Interpret the statements \(f(100)=16\) and \(f^{\prime}(100)=.25.\) (b) Estimate the value of one share on the 101 st day since the company went public.

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=\frac{1}{\sqrt{x}}$$

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=x^{2}$$

(a) Draw two graphs of your choice that represent a function \(y=f(x)\) and its vertical shift \(y=f(x)+3.\) (b) Pick a value of \(x\) and consider the points \((x, f(x))\) and \((x, f(x)+3) .\) Draw the tangent lines to the curves at these points and describe what you observe about the tangent lines. (c) Based on your observation in part (b), explain why $$\frac{d}{d x} f(x)=\frac{d}{d x}(f(x)+3)$$

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\frac{1}{x}$$
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